Pyramids and monomial blowing-ups

2004/09/23 by M. J. Soto, Soto, M. J., José L. Vicente +1
Mathematics · #13H99 (Secondary) #15A39 (Primary) #Commutative Algebra (math.AC) #FOS: Mathematics #Optimization and Control (math.OC) #math.AC #math.OC #msc:13H99 #msc:15A39

paper · pdf · doi:10.48550/arxiv.math/0409446

12 pages

arxiv created 2004/09/23 · arxiv updated 2009/12/01

Abstract

We show that a convex pyramid in Rⁿ with apex at 0 can be brought to the first quadrant by a finite sequence of monomial blowing-ups if and only if its intersection with the opposite of the first quadrant is 0. The proof is non-trivially derived from the theorem of Farkas-Minkowski. Then, we apply this theorem to show how the Newton diagrams of the roots of any Weierstrass polynomial are contained in a pyramid of this type. Finally, if n = 2, this fact is equivalent to the Jung-Abhyankar theorem.

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